If you are solving an elliptic equation for the temperature with linear shape functions, the resulting discretization should be equal to that obtained with a second order FDM and the 4 nodes at the corners are not involved.

Thus, check if they really enter into the system.

There's no established rule for this. The best option is to use 50°C since it is the largest.

Thank you Chennakesava Kadapa

I was also thinking the same, my logic was, that the largest will fulfill all other specified value at the common node.

Dear Dr. Filippo Maria Denaro ,

Thank you for your response. However, I am new to FEM and hard time understanding your comment.

However, the corner is a singularity and as such must be treated. and a doubled -value node can be also used.

Hi everyone,

I think that you try to solve a problem that is not well posed as written here. I mean that in the Finite Element (FE) theory and for the heat equation, the boundary conditions, especially Dirichlet conditions, must be established at mesh nodes. This is a prerequisite to stay in the "expected" Lagrange space associated to the Hilbert space where Lax Milgram theorem must be applied to state existence and uniqueness of the solution. Hiroaki Nishikawa proposes to define boundary conditions at the mesh faces. Such a procedure is from the standard finite volume formulation. Even if the duality of FE and FV over dual cells establishes a bridge between formulations, the proposed question is not so clear. In my opinion, introducing multi-valued points in the discretization is an artefact to circumvent the limitation and in addition, it can introduce some requirements on the mesh itself (edge swap for instance). Do you agree with me ?

There is simple solution to this - you need to use cell centered discrete action for temperature, Then Boundary conditions will be defined at the center of boundary faces. There is no unknowns or know temperatures at the nodes.

I mean cell-centered discretization. Look at mimetic discretizations.

Chennakesava Kadapa Filippo Maria Denaro Hiroaki Nishikawa

I have a doubt.

If I have three adjacent planes with different boundary conditions, in a 3D domain, Dirichlet (fixed temperature) on one plane, Neumann fixed flux on another plane and Neumann heat conduction on another, then what should I do?

Should I consider all the conditions on the common node? I read somewhere that if Temperature and heat flux is specified on a node then only specified temperature should be considered, but I don't know if I should ignore convective heat transfer when temperature is specified.

And in 2D case, when only temperature is specified on one edge, and convective heat transfer on adjacent edge? Then should I consider the heat convection at the common node these two edges?

Thank you 😊